Embedded localization method

On the other hand, some other alternative approaches have been developed to model the fracture as the XFEM [194] or the embedded localization method [195], These approaches can also be combined with a cohesive zone model [196]. 

An alternate approach, which has proven to be extremely useful for metals, has been developed by Daw, Baskes and Foiles - (and to a lesser extent, by Ercolessi, Tosatti and Parrinello ). Called the embedded atom method (EAM) (or the glue model by the second group), the interactions in this approach are developed by considering the contribution of each individual atom to the local electron density, and then empirically determining an energy functional for each atom which depends on the electron density. This circumvents the problem of defining a global volume-dependent electron density. 

While the embedded atom method has been formally derived by Daw and Baskes the functions used in computer simulations are t3pically empirically determined. The description presented here will therefore treat this approach as an empirical method. The first step in determining the potential is to define a local electron density at each atomic site in the solid. A simple sum of atomic electron densities has proven to be adequate, and so in most cases a sum of free atom densities is used . The second step is to determine an embedding... 

Assume that the binding curves for the asymptotic fragments are available and have been used to determine the relevant parameters (7>ab> ab> ab)> (Das, Kas. l As) (Dj,s, bs. bs)- The parameters for the interaction with jellium, (Dah, ah. ah) and (Dbh, bh. bh). are not adjustable, being determined by the SCF-LD values. Thus, the remaining variables in the PES, the so-called Sato parameters, Aab>A s, and A s, are undetermined and available for flexible representation of the full molecule-solid reactive PES. We consider the effect of these on the PES later. At this point, we do want to emphasize that the basic physics and chemistry— (1) interactions with localized and delocalized metal electrons and (2) nonadditive chemical bonding—are correct. We should also note that the representation of an interaction in metals in terms of an embedding function (in jellium) plus two-body terms is identical in spirit to the embedded atom method (EAM) (Daw and Baskes 1984, 1988). The distinction here is that we do not use the EAM for the A-B interaction and explicitly incorporated nonadditive energies via the LEPS prescription, both of which are important for the accurate representation of the reactive PES. 

Chapter 11 deals with the tight-binding and the embedded-atom models of solid state. The method of the local combination of atomic orbitals is described. We present examples of the technique application. Description of atom systems in the embedded-atom method, embedding functions and applications are considered. In conclusion the reader will find the review of interatomic pair potentials. 

The embedded-atom method (EAM) overcomes the limitations of the pair potential technique. It is considered to be practical enough for calculations of defects, impurities, fractures, and surfaces in metals. In this model, an impurity (that is, a quasi-atom) is assumed to experience a locally uniform or only slightly nonuni-form, environment. The energy of the quasi-atom can be expressed as... 

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

In the case of localized states related to the dopants—including transition-metal (TM) and rare-earth (RE) ions—the effect of hydrostatic pressure on the local energetic stmcture and electronic transitions is simulated by the reduction of the size of the cluster, which includes the dopant ion and the ligands. The influence of pressure on the energetic stmcture of the TM and RE ions can be simulated using crystal-field phenomenological model calculations [10-12]. However, a more advanced approach, i.e., ah initio model potential embedded-cluster method, also has been used [13]. 

Several studies have focused on extensive MD simulations of Pt nanoparticles adsorbed on carbon in the presence or absence of ionomers [109-113]. Lamas and Balbuena performed classical molecular dynamics simulations on a simple model for the interface between graphite-supported Pt nanoparticles and hydrated Nation [113]. In MD studies of CLs, the equilibrium shape and structure of Pt clusters are usually simulated using the embedded atom method (EAM). Semi-empirical potentials such as the many-body Sutton-Chen potential (SC) [114] are popular choices for the close-packed metal clusters. Such potential models include the effect of the local electron density to account for many-body terms. The SC potential for Pt-Pt and Pt-C interactions provides a reasonable description of the properties of small Pt clusters. The potential energy in the SC potential is expressed by... 

At present the best method for calculation is the ab-initio molecular-dynamics method allowing simultaneous calculation of the evolution of the atomic system and electron subsystem. In this chapter, however, the classical molecular d5Uiamics method in combination with semiempirical potentials of atomic interaction is used in the fiamework of the embedded-atom method (EAM) [10]. On the one hand, the EAM-approach proved to be good for the simulation of the metal atomic stmcture in crystalline and liquid states. On the other hand, the EAM-approach is a reasonable compromise between the calculation complexity and physical validity, which allows to conduct the simulation of a system consisting of a larger munber of atoms than that in Refs. [6-9]. In addition, it will allow to establish to what extent the results of the local cluster stmcture simulation are sensitive to the model describing interatomic bonds. 

Since the pioneering cluster calculation on the KNiFs solid of Shulman and Sugano [6] there has been a wide variety of proposals of procedures to handle relatively localized electronic states of a solid with a molecule-like Hamiltonian that includes the relevant solid host effects, depending on the type of solids and on methodological flavors (Green s functions, wavefunctions, density functional, etc.). A recent summary of practical methods can be found in Huang and Carter [7]. Here we describe our choice of embedded-cluster method, particularly useful in ionic materials. 

A first step towards a systematic improvement over DFT in a local region is the method of Aberenkov et al [189]. who calculated a correlated wavefiinction embedded in a DFT host. However, this is achieved using an analytic embedding potential fiinction fitted to DFT results on an indented crystal. One must be cautious using a bare indented crystal to represent the surroundings, since the density at the surface of the indented crystal will have inappropriate Friedel oscillations inside and decay behaviour at the indented surface not present in the real crystal. 

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